New constraints on coherent elastic neutrino–nucleus scattering by the νGeN experiment

The investigation of neutrino properties is a fast-developing area of modern particle physics. Recently, significant advances have been achieved in the search for and study of coherent elastic neutrino–nucleus scattering, a process predicted by the Standard Model [1, 2]. Due to a small momentum transfer, the neutrino interacts simultaneously with all nucleons, and the cross-section of such a process is enhanced by several orders of magnitude compared with those in other neutrino interactions at the same energy [3, 4]. The differential cross-section of CEνNS for a spin-zero nucleus with a mass M can be expressed as [5]

where T is the nuclear recoil energy, $ E_{\nu} $ is the neutrino energy, Q is the transferred momentum, and$ F(Q^{2}) $ is the nuclear form-factor. The Fermi constant is labeled as $ G_{F} $, and $Q_W = N - (1-4 \sin^2 \theta_W) Z$ is the weak charge of a nucleus. Since the predicted value of the Weinberg angle at low energies is $\sin^2 \theta_W = 0.23867\pm0.00016$ [6], the full CEνNS cross-section σ is almost proportional to $ N^2 $, the square of the number of nuclear target neutrons. Studying CEνNS allows us to test the Standard Model, search for non-standard neutrino interactions, probe some aspects of nuclear physics, and perform other investigations [7].

The first observation of CEνNS [8] using neutrinos produced by the SNS accelerator [9] was reported by the COHERENT experiment. Researchers are actively investigating the coherent elastic scattering of reactor antineutrinos off nuclei, with many experiments underway or under construction [10−17]. The sensitivity of these experiments is approaching the CEνNS detection. The search for CEνNS at reactors is of particular interest because it offers opportunities to probe physics beyond the Standard Model and to monitor reactor operations. Notably, it enables the detection of neutrinos below the 1.8 MeV energy threshold required for inverse beta decay [18]. Due to a higher neutrino cross-section, the detectors typically have a much smaller size compared to detectors that register reactor antineutrinos through inverse beta decay.

A high neutrino flux, low background, large target mass, and low energy threshold are required for CEνNS detection. The signals of interest can be obscured by backgrounds such as elastic neutron scattering or electronic noise, which complicates detection. The cosmogenic background poses a significant challenge for CEνNS experiments with a shallow overburden because it is difficult to supress due to the production of secondary fast neutrons within the setup shielding [19]. The νGeN experiment has one of the best locations to search for CEνNS – the Kalinin Nuclear Power Plant (KNPP) in Udomlya, Russia, Unit #3, near a WWER-1000 type 3.1 GW$ _{\rm th} $ reactor [20]. The close vicinity of the reactor core (11.1 m) enables the utiliziation of one of the largest possible fluxes of antineutrinos of up to $ \sim $4.4$ \times 10^{13} $ $ \mathrm{cm^{-2} s^{-1}} $. The scheme of the reactor site and of the setup location (Room A-336 [21]) are shown in Fig. 1. The experimental site is just under the reactor core, which together with the other construction materials of the reactor provides shielding from cosmic rays of approximately 50 m w.e. Such overburden allows us to completely remove the hadronic component of the cosmic flux [22]. The muon flux suppression factor measured at a similar KNPP reactor unit is up to 13 times, depending on the zenith angle [23]. Therefore, the background from secondary neutrons is significantly less than that of a shallow laboratory. The room is 8.7 by 9.3 m and has a height of 4.1 m. The experimental setup was at the centre of the room on a lifting device, which enabled the adjustment of the distance to the center of the reactor core from 12.5 to 11.1 m. These distances correspond to neutrino flux changes from$ 3.4 \times 10^{13}$ to $ 4.4 \times 10^{13} $ $ \mathrm{cm^{-2} s^{-1}} $based on the calculation method from [24].

Figure 1.  (color online) Scheme of the reactor unit and the site of the νGeN experimental setup (not to scale).

The νGeN experiment uses a custom-designed high-purity germanium (HPGe) detector manufactured by Mirion Technologies (Canberra Lingolsheim) [25]. The p-type germanium crystal has a cylindrical shape with a diameter of 70 mm and a height of 70 mm. The sensitive volume of the detector is 265 cm$ ^3 $, which corresponds to an active mass of 1.41 kg. The detector is installed inside a cryostat made mostly of low-background aluminum and copper. The photo of the germanium detector before installation is shown in Fig. 2. The germanium crystal is cooled by an electrically powered pulse tube cooler (Cryo-Pulse 5 Plus (CP5+)) [26]. The cooling temperature of the detector is set to −185 ℃. This temperature is optimal for minimizing noise and improving the energy resolution of the current detector. The cooling power depends on the ambient temperature and is typically approximately 80 W. The detector is equipped with a 60-cm-long cooling rod allowing for sufficient shielding from external radiation.

Figure 2.  (color online) νGeN detector before deployment at KNPP.

The innermost part of the shielding is made of 3D-printed nylon, which displaces air and reduces potential radon contamination. The layer density was set to 15% of nylon density to ensure that the layer is soft enough to prevent cryostat damage during the installation of the shielding. The thickness of the nylon layer is 16−46 mm, depending on the direction. The subsequent shielding layers consist of 10 cm of oxygen-free copper, 8 cm of 3.5% borated polyethylene, 10 cm of lead, an additional 8 cm of borated polyethylene, and a 5-cm-thick active muon veto composed of plastic scintillator panels. The radon level inside the shielding is further decreased through nitrogen expulsion. The cryocooler of the detector is placed on an dynamic antivibration platform, TS-C30 [27], to minimize vibrations from surrounding equipment. The scheme of the setup structure including the passive and active shielding is shown in Fig. 3.

Figure 3.  (color online) Scheme of the νGeN shielding (top view, not to scale).

The germanium diode is electrically depleted by a positive operating voltage of 2300 V produced by a CAEN power supply, Mod. No. 1471HA. Ionization energy losses induced by incoming particles passing through the HPGe detector result in a charge being collected on the electrodes. The charge is converted into voltage-amplitude pulses by integrated cold and warm electronics. The detector is equipped with a charge-sensitive preamplifier with pulsed-reset feedback, contributing to the reduction of the noise of the feedback resistor [28]. The preamplifier requires periodic baseline resets once the dynamic range becomes saturated. This behavior is illustrated in the oscilloscope screenshot (see Fig. 4).

Figure 4.  (color online) Screenshot from the digital oscilloscope demonstrates a typical output from the preamplifier of the HPGe detector under laboratory conditions without the described shielding.

The dynamic range of the preamplifier is up to approximately 7 V, which is equivalent to approximately 2.3 MeV in the energy scale. A relatively large signal immediately initiates the reset of the baseline and cannot be detected. The reset frequency depends on the sum of the detector leakage current and count rate. For the νGeN detector inside the shielding at KNPP, the reset rate is approximately 5–10 Hz. The preamplifier features two similar amplitude outputs (OUT E and OUT E2), an inhibit output, and a test signal input. The inhibit output provides a logic signal during the reset period. The duration of the inhibit signal is manually set to 800 μs to exclude artificial signals generated by the reset. The signals from the output are shaped and amplified to obtain a positive signal suitable for the analog-to-digital converter (ADC). The parameters of each event (energy, timestamps) are evaluated by the multichannel analog-to-digital converter CAEN VME Realtime ADC V785N.

Each output of the preamplifier is connected to two ORTEC 672 spectroscopic amplifiers, resulting in four amplifiers in total. The output signals from these amplifiers are processed by an ADC, producing four reconstructed energy values corresponding to channels 0 through 3. Comparing and averaging the reconstructed signals from the different preamplifier outputs helps suppress electronic noise, thereby improving energy resolution and lowering the energy threshold. Figure 5 shows a diagram of the components involved in the data acquisition. A wide energy range of up to 700 keV is measured with one of the ORTEC 672 amplifiers (labeled HE in Fig. 5). Timestamps from the muon veto system and inhibit signals are processed by the same ADC. The other channels are tuned up for measurements below $ \sim $13 keV. The CAEN VME V976 trigger unit issues an acquisition command on input conditions corresponding to 1) a low-energy HPGe signal, 2) a high-energy HPGe signal, 3) an inhibit logical signal, and 4) a muon veto.

Figure 5.  (color online) Diagram of the acquisition system of the νGeN experiment.

The acquisition software of νGeN is based on the acquisition software previously designed for the DANSS experiment [29, 30]. It records real-time information about each event, including the amplitudes of all channels and the time of the event. Owing to KNPP safety restrictions, the experimental hall does not have internet access. Hence, the data are copied by shifters once per week for offline analysis. The shifters also examine all equipment ensuring its stable and correct performance, and typically restart a new run every week or, if needed, more often. The offline analysis is performed later with ROOT software [31].

The cooling power of the CP5+ depends on the room temperature; therefore, an increase in temperature would increase the cooling power of the CP5+ and may lead to more noise in the detector. The experimental hall is equipped with air conditioners to decrease the temperature in the room and provide stable ambient conditions. The temperature and humidity in the room are continuously recorded by two devices located in different locations in the room. Typically, the temperature is approximately 22 ℃, stabilized within $ \pm $1 ℃. The neutron background outside the shielding is controlled with a low-background neutron detector based on the CHM-57 counter [32], which was developed at JINR and used in a few experiments [11, 23].

The high-energy part of the spectrum is calibrated with a few-gram piece of a tungsten welding rod that contains approximately 2% of $ ^{232} $Th. The energy calibration of the low-energy part of the spectrum is determined by means of the 10.37 keV cosmogenic line of $ ^{68,71} $Ge and artificial lines created by the pulse generators CAEN Mod. NTD6800D and ORTEC 419, which help to establish the calibration line slope. The shape of the generated pulses was rectangular, providing a response similar to the response of the physical pulse from the detector. The calibration of the low-energy part of the spectrum is verified by checking the position of the 1.3 keV cosmogenic line of the EC decay of $ ^{68,71} $Ge. Typically, the calibration with the thorium source and pulse generator is performed monthly or after any change of the experimental conditions.

The goals of the experiment require the best possible energy resolution and a low energy threshold connected to it. Preliminary measurements showed that the lowest threshold and the best resolution are achieved with the 6 μs shaping time. The energy resolution obtained with the pulse generator is 101.6$ \pm $0.5 eV (FWHM). To reduce the impact of noise in the electronic chain, the final signal energy is calculated as a weighted combination of two reconstructed energies from different preamplifier outputs, with weights of 2/3 and 1/3 for channels 1 and 0, respectively. This weighted averaging of reconstructed energies was found to reduce noise events and lower the energy threshold. In addition to averaging, consistency between energy reconstructions from the two different channels can be checked. Differences in energy reconstruction indicate the influence of electronic tract noise, which may lead to artificial events. Figure 6 shows the two dimensional histograms of the events reconstructed using different preamplifier chains. As shown in Fig. 6, the events from the pulse generator are located mostly diagonally, similar to the physical events. Through graphical cuts, which exclude events outside the diagonal area, nonphysical noise events generated in the electronic chain can be removed from the data.

Figure 6.  (color online) Two-dimensional energy histograms from two identical channels for pulse generator (left) and background events (right). The solid lines are graphical cuts that remove noise events.

In addition, the comparison of the signals obtained with different shaping times allows for efficient noise discrimination [33, 34]. We used shaping times of 6 and 10 μs to compare the signals. Similar to Fig. 6, graphical cuts are used to suppress events originating from noise based on the differences in energy reconstruction with varying shaping times. The discrimination parameters for these graphical cuts are shown in Fig. 7.

Figure 7.  (color online) Two-dimensional energy histograms from two channels with different shaping times for pulse generator (left) and background events (right). The solid lines are graphical cuts that exclude noise events.

The efficiency of these cuts was calculated using measurements from the pulse generator. By comparing the observed intensity of the peaks in the energy spectrum with the pulse rate from the generator, we can determine the efficiency of both the graphical cuts and the trigger. The results of these investigations are shown in Fig. 8. The detection efficiency was higher than 90% for signals above 0.3 keV. To fit the efficiency distribution, we used the following function F:

Figure 8.  (color online) Efficiency of signal detection measured with the pulse generator.

where E is the energy, and $ a,b,c,d,f $ are the free parameters. The parameter values maximizing the likelihood of the fit are$ a = 0.2006\pm0.0022 $, $ b = 0.0821\pm0.0016 $, $ c = -0.0177\pm0.0033 $, $ d = -2.07\pm0.21 $, and $f = 0.42\pm 0.09$. The obtained spectra are corrected according to the evaluated efficiency function.

The reset of the baseline produces small afterpulses, which can also be interpreted as physical signals. Nonphysical events induced by the reset are clearly visible in Fig. 9, demonstrating the correlation between the energy of the event and delay after the inhibit signal. To exclude these artificial events, the time period of 4.8 ms after each of the resets is not considered in the analysis.

Figure 9.  (color online) Distribution of time to the previous inhibit signals versus energy, demonstrating events connected to the reset.

The investigation of the time difference between consecutive events reveals another nonphysical population of signals, likely caused by resets and microphonic noises. These noise events can be mitigated by excluding any of the consecutive events within 150 μs.

The energy spectrum, both before and after applying the quality cuts and the anti-coincidence requirement with the muon veto, is shown in Fig. 10. The artificial peak at approximately 0.8 keV, caused by the reset, is removed after applying the inhibit cut. Both the muon and inhibit cuts remove some physical events due to the dead time introduced by the muon or inhibit signals. A correction for this dead time, calculated to be 9.0%, was applied to the obtained spectrum. Additionally, the efficiency of the consecutive time cut was evaluated by comparing the count rates before and after its application, revealing a dead time of approximately 0.1% from the consecutive time cut.

Figure 10.  (color online) Energy spectrum before (black) and after the application of several cuts: inhibit+time cuts (blue), further application of muon veto and graphical cuts (green), and after removing time intervals with high noise levels (red) (see section IV for details).

The stability of data collection is crucial for accurate data interpretation. A direct comparison between ON and OFF datasets is not justified in the CEνNS search if background and noise count rate fluctuations are comparable to those of the expected signal. This section discusses factors that may cause undesirable variations and outlines the selections made to ensure stability within the CEνNS dataset collected from September 2022 to May 2023, at a distance of 11.1 m from the reactor core.

First, we consider the influence of $ ^{222} $Rn-related backgrounds on the CEνNS region of interest (ROI). The activity of radon inside the setup shielding was characterized on the basis of the 609 keV line from $ ^{214} $Bi. The stability of the corresponding rate in time and of accumulated counts is shown in the histograms in Fig. 11. Data acquired in September 2022 were excluded from the dataset owing to increased radon activity during this period. The remaining data suggest a stable radon concentration with no difference between ON and OFF datasets within statistical uncertainty.

Figure 11.  (color online) Top: count rate of the 609 keV $ ^{214} $Bi line vs. time for ON (red) and OFF (black) data. The dataset to the right of the dashed line is selected for further analysis. The ON and OFF periods overlap within 10-day intervals (see also Fig. 12). Bottom: accumulated count rate at 609 keV for ON (red) and OFF (black) data after selection.

The germanium detector was delivered to KNPP in 2019 and has been stored inside the shielding since November 2019. The intensity of the cosmogenic 10.37 keV line during reactor ON and OFF periods was found to be 14.21$ \pm $0.31 and 14.4$ \pm $0.6 counts/(kg d), respectively. Thus, no significant decrease in the gamma background was observed during the selected analysis period.

We ensured that the fluctuations of low-energy noise have a negligible effect on the count rate in the ROI of CEνNS. Hence, we excluded data of periods when room temperature changes exceeded the mean by $ \pm $1 ℃. Temperature variations can be connected to noise levels owing to changes in the cryocooler power and corresponding mechanical vibrations. We also excluded the first 30 min of the data of a new run along with the final 10 min of the run to avoid possible noise produced by on-site personnel. Noise events can significantly influence the energy spectrum. To exclude these events, we used an energy threshold of approximately 0.29 keV above the region dominated by noise. Subsequently, we evaluated the stability of the count rate in the energy range from 0.25 to 0.28 keV, well below our analysis region and dominated by noise (see Fig. 12). One of the two channels (Channel 1 – black dots) showed a significant time dependence. Although we did not observe any significant changes in the rates for the ROI, the time periods with count rates $ > $ 0.01 in Channel 1 were excluded from further analysis to avoid possible noise contamination on the spectrum.

Figure 12.  (color online) Stability of the count rates observed in the noise energy region from 0.25 to 0.28 keV and reactor thermal power in time.

After excluding noisy time periods, we verified that residual noise fluctuations do not affect the ROI count rate by examining its stability near the detection threshold. In particular, we fitted the dependence of the count rate on time to the constant and determined the p-values ¹ of the $ \chi^2 $-score for ON and OFF data separately. The count rate became sufficiently stable for the threshold of 0.29 keV. The p-values of 94% and 9% were derived for ON and OFF data, respectively, in the energy interval from 0.29 to 0.31 keV. Therefore, no statistically significant deviation from the constant count rate was observed. The stability achieved through the above selections is confirmed by the count rate observed in the full CEνNS ROI, ranging from 0.29 to 0.4 keV. Its dependence on time agrees with the constant for ON and OFF datasets analyzed separately; see Fig. 13. The corresponding p-values are 82% and 10% for the ON and OFF data, respectively. After applying all stability selections, 194.5 and 54.6 kg$ \cdot $days of data remained in the analysis for the reactor-ON and reactor-OFF periods, respectively.

Figure 13.  (color online) Stability of the count rate in CEνNS ROI from 0.29 to 0.4 keV for ON (red) and OFF (black) periods. The energy-dependent selection efficiency constant is not corrected in time (see Fig. 8).

The calculation of the expected nuclear recoil spectrum from CEνNS accounts for the antineutrino energy distribution and the detector parameters. Information on the reactor's isotope fraction and thermal power was provided by KNPP personnel. The average fission fractions of the main isotopes of $ ^{235} $U, $ ^{238} $U, $ ^{239} $Pu, and $ ^{241} $Pu were 0.642, 0.070, 0.246, and 0.042, respectively, for the analyzed period of the reactor-ON data. The calculation [35] based on this fuel composition predicts a thermal energy of 204.7 MeV per fission. The average thermal power obtained for the reactor-ON dataset was 3081 MW. The reactor antineutrino energy spectra of up to 11 MeV were calculated using the summation model [36] based on the parameters described above. This antineutrino spectrum was used to calculate the CEνNS nuclear recoil energy distributions for each germanium isotope. The recoil spectra of five stable isotopes produce a summation spectrum.

The CEνNS nuclear recoil spectrum must be corrected to account for the quenching of ionization signals from nuclear recoils, typically described by the Lindhard model [37]. However, recent measurements at low nuclear recoil energies show significant discrepancies [38−41] introducing uncertainty in the predicted strength of the CEνNS signal (see Fig. 14). In this study, we consider three models for the quenching factor (QF) dependence on energy. The first is the Lindhard model ($ k=0.162 $), suggested by the measurements from CONUS [39] (further referred to as "C"). The decreasing trend of QF with the decrease of nuclear recoil energy is supported by Ref. [40], although with somewhat lower QF values. The second model is suggested by the "iron filter" data of the Dresden-II experiment [38]. This model (further referred to as "$ D1 $") assumes a linear fit of corresponding data below 1.35 keV$ _{nr} $ combined with the Lindhard model ($ k=0.157 $) above that. The increase of QF with the decrease of energy is supported by Ref. [41], which provides data for 254 keV nuclear recoils only. The latter demonstrate a QF lower than the one expected from the linear fit of $ D1 $ (25% vs. 38% respectively). Finally, we consider the intermediate QF based on "photo-neutron" data of Ref. [38]. This model is further referred to as "$ D2 $" and is represented by the curve from the supplemental materials to Ref. [14]. The scenarios we consider illustrate the spread of existing experimental data. The first one (C) leads to a lower reactor CEνNS count rate, while the second ($ D1 $) and the third ($ D2 $) demonstrate a more optimistic CEνNS prediction. The resulting nuclear recoil spectra expected from CEνNS in the νGeN detector for these QF models are shown in Fig. 15.

Figure 14.  (color online) Measurements of germanium nuclear recoil QF [28, 38−46], the photo-neutron data from Ref. [38] are not shown for clarity but are represented by one of the solid lines. Solid lines: C (red), $ D1 $ (magenta) and $ D2 $ (green) models, see text for details.

Figure 15.  (color online) Expected count rates and energy spectra of CEνNS in the νGeN setup.

The experimental energy deposition spectra acquired during reactor-ON and reactor-OFF periods are shown in Fig. 16. To obtain these spectra, no scaling factors, besides the correction to the selection efficiency and live time, were used.

Figure 16.  (color online) Comparison of the spectra acquired when the reactor is ON (red) and OFF (black).

We evaluated constraints on the CEνNS amplitude based on the statistical analysis of the residual ON$ - $OFF spectrum in ROI from 0.29 to 0.40 keV. The lower boundary of the ROI was chosen to eliminate potential noise contributions, while the upper boundary was determined through analysis of simulated spectra to maximize sensitivity to CEνNS given the current background level. The number of counts in each bin of both the ON and OFF spectra is sufficient to ensure that the statistical uncertainty in the residual spectrum follows a Gaussian distribution. We fitted the residual spectrum using a CEνNS prediction shape that minimizes $ \chi^2 $ statistics. The only free parameter in the fit is the CEνNS amplitude A in units of "times Standard Model prediction" ($ \times $SM). The results of the fit for the CEνNS predictions based on the C, $ D1 $, and $ D2 $ quenching models are shown in Fig. 17 (top) with predictions presented for a comparison (bottom).

Figure 17.  (color online) The residual ON$ - $OFF spectrum, the best signal shape fits (top) and the nominal signal predictions (bottom) for C (red), $ D1 $ (magenta) and $ D2 $ (green) QF.

Table 1 presents the best fit values $ A_{\rm best} $ and their statistical uncertainties $ \sigma_{A} $ as well as the limits at a 90% confidence level (CL) on the CEνNS amplitude. We also cite the value of the sensitivity, i.e. the expected median limit evaluated using the OFF data only. The $ \Delta\chi^2 $ profiles corresponding to the data fit are shown in Fig. 18. The best fits under the assumption of C and $ D2 $QF do not contradict the Standard Model CEνNS and the null hypothesis. The 90%-CL upper limits on the CEνNS amplitude are approximately 4.3/3.1 times larger than the Standard Model prediction, respectively.

QF	$A_{\rm best}\pm \sigma_{A}$, $ \times $SM	$\chi^2_{\rm best}$ (ndf=10)	S,$ \times $SM	L,$ \times $SM
C	$ 1.5 \pm 1.7 $	13.6	3.8	4.3
D1	$ 0.1 \pm 0.4 $	14.4	1.6	0.7
D2	$ 0.8 \pm 1.4 $	14.1	3.3	3.1

Table 1.  Results of the statistical analysis of the residual spectrum, along with sensitivity (S) of the experiment and CEνNS amplitude limits (L) at 90% CL.

Figure 18.  (color online) The profiles of $ \Delta\chi^2 $ statistics for the fit of the residual spectrum to the CEνNS prediction shapes: C (red), $ D1 $ (magenta), and $ D2 $ (green) QF.

The fit for the most optimistic $ D1 $ QF model suggests the exclusion of the Standard Model CEνNS rate at $ 2.5\sigma $. Such a result indicates either the presence of non-standard neutrino-quark interactions (NSI) suppressing CEνNS or a bias in the QF estimate. We confirm the earlier result of the CONUS experiment [47] excluding the Standard Model CEνNS for the $ D1 $ quenching scenario. Our result is also in tension with the reactor CEνNS detection claim of the Dresden-II experiment [14]. We note that the strength of this tension is challenged by one of the systematic uncertainties described below. For completeness of discussion, the recent $ 3.9\sigma $CEνNS detection on Ge by COHERENT should be mentioned [48]. The observed CEνNS rate at the nuclear recoil energy range not affected by the QF discrepancy is approximately $ 2\sigma $ lower than the Standard Model prediction.

In what follows, we discuss the sources of systematic uncertainties and their impact on the result. The largest uncertainty is related to the nuclear recoil QF model. Resolving the discrepancy between the results of Ref. [39] and Ref. [38] requires more experimental data for nuclear recoil energy below 1.3 keV. The reactor measurements of CEvNS can provide another way to test QF, although the NSI and incorrect QF scenarios are hard to distinguish. Both the CEνNS signal amplitude and QF-related parameter can be treated as free parameters in the fit and evaluated with the improved energy threshold and increased statistics expected in the CONUS+ [49], TEXONO [50], Dresden-II, νGeN, and future germanium experiments. It is hard to say if this approach is feasible due to significant degeneracy between the CEνNS rate and QF. The Ge-based bolometers can test the CEνNS amplitude independently from the ionization quenching if only the thermal signal is considered.

The second largest systematic uncertainty is associated with the energy scale calibration. To estimate this uncertainty, we use two modifications to the calibration procedure. The first method ("global’") utilizes the full dataset without considering minor variations observed in regular calibrations, offering higher statistical precision for the 10.37 keV calibration line. The second method ("modified") accounts for potential inaccuracies in energy reconstruction at low energies by applying a slightly adjusted calibration slope, resulting in an estimated shift of approximately 15 eV toward higher energies within the ROI. Table 2 demonstrates possible changes in the results due to the modification of the calibration energy scale. No significant effect on the final results of QF in the cases of C and $ D2 $. For the $ D1 $ scenario, the tension with the Standard Model CEνNS decreased from $ 2.5\sigma $ to a modest $ 1.6\sigma $, below the 90% CL.

Energy scale	$A_{\rm best}\pm \sigma_{A}$ (C/D1/D2)	Limit (C/D1/D2)
Default	$ 1.5 \pm 1.7 $ / $ 0.1 \pm 0.4 $ / $ 0.8\pm 1.4 $	4.3 / 0.7 / 3.1
Global	$ 1.8 \pm 1.7 $ / $ 0.1 \pm 0.4 $ / $ 1.0\pm 1.4 $	4.5 / 0.7 / 3.3
Modified	$ 1.2 \pm 2.4 $ / $ 0.0 \pm 0.6 $ / $ 0.6\pm 2.1 $	5.1 / 1.1 / 4.1

Table 2.  Effect of the systematic uncertainty of the energy calibration scale on results with different QFs. The best fit values ($A_{\rm best}$) and limits (90% CL) are in units of $ \times $SM.

Another systematic uncertainty is associated with the reactor antineutrino energy spectrum. The CEνNS count rate expected above the detector threshold is affected by the presence and magnitude of the high-energy part ($ E_{\nu}>8 $MeV) of the antineutrino flux. In this study, we test two of the available spectra models: the Summation Model 2018 [36] (SM2018) and the model developed by authors from the Institute for Nuclear Research of the Russian Academy of Sciences in Ref. [51] (INR RAS) verified using the data from the Double Chooz experiment [52]. Both models were implemented with average fission fractions of the νGeN exposition considered in this work. The results of these implementations are compared to each other and to the antineutrino energy spectrum measured by the Daya Bay experiment (DB) [53, 54] in Fig. 19. The count rate predicted under the assumption of INR RAS is smaller than that for SM2018 by 10%−15% in the νGeN CEνNS ROI. The DB-based estimate exceeds that for SM2018 by approximately 3%. The effective fuel composition of DB is 56.4% ($ ^{235} $U), 7.6% ($ ^{238} $U), 30.4% ($ ^{239} $Pu), and 5.6% ($ ^{241} $Pu), slightly different from the average composition for the νGeN exposition. The estimates show that, for the same fuel composition, the discrepancy between SM2018 and DB increases up to approximately 5%. Thus, the non-negligible spread in the expected CEνNS count rate should be considered as the systematic uncertainty of the results.

Figure 19.  (color online) Effect of reactor antineutrino spectra models on the CEνNS count rate estimates in the ROI of νGeN.

To date, CEνNS has been observed at the SNS using a CsI[Na] scintillator [55], liquid argon [56], and germanium [48]. Recent results of experiments on the direct search for dark matter [57, 58] indicate the coherent scattering of solar boron neutrinos off xenon nuclei. The debated claim of CEνNS detection at reactors [14, 59] and the most stringent upper limit (aapproximately twice the Standard Model prediction assuming the C-model QF) [47] originate from two germanium-based experiments: Dresden-II and CONUS ², respectively. These two results, however, are in tension with each other under the assumption of the $ D1 $-model QF. Given this tension and a 2-σ deviation from the SM for the result obtained in the COHERENT experiment [48], more data are required for the conclusive observation of CEνNS at reactors, along with measurements of the CEνNS cross-section on germanium. Hence, the potential of the νGeN experiment is estimated below.

The typical reactor operation schedule at KNPP includes a 45 day OFF period for every 16.5 months of ON time. Under this schedule, the statistical uncertainty in the residual count rate is primarily governed by the OFF data. Using the background spectrum measured during reactor shutdown, we extrapolate the sensitivity of νGeN for larger expositions, considering both the C and $ D2 $QF models. The significance of the expected null hypothesis rejection as a function of data taking time is shown in Fig. 20. The "time" along the horizontal axis in Fig. 20 corresponds to the accumulated OFF data (and assumes 11 times more ON than OFF at any moment). In this scenario, the 3-σ null-rejection can be achieved only for the exposition longer than 27/19 years for $ C/D2 $ QF assuming an energy thrshold of 0.29 keV. Another approach assumes using a background model instead of the OFF data [19]. If the systematic uncertainty of the model based on background decomposition is negligible compared to the statistical uncertainty of the ON spectrum, then the "time" axis in Fig. 20 corresponds to the ON exposition. In this case, the 3-σ level can be achieved within a few years.

Figure 20.  (color online) Projected sensitivity of νGeN based on the background count rate measured at KNPP for C (red) and $ D2 $ (green) QF scenarios. The "time" on the X-axis corresponds to the OFF or ON statistics depending on the analysis strategy (see text for details).

Currently, we consider the decomposition of the νGeN background measured at KNPP, as well as its reduction by the deployment of the NaI-based "Compton Veto" within the setup shielding. Additional laboratory tests are performed to improve the energy threshold of the HPGe detector by decreasing the power consumed by the cryocooler and extracting the waveforms from the detector for the offline pulse shape discrimination of surface events and noise [61].

We considered the νGeN HPGe detector dataset, including 194.5 kg$ \cdot $days of reactor operation and 54.6 kg$ \cdot $days of reactor shut down, to search for signals from coherent scattering of antineutrinos off germanium nuclei. The results of the fit of the signal prediction to the residual ON$ - $OFF spectrum in the interval from 0.29 to 0.40 keV do not contradict both the null hypothesis and the Standard-Model CEνNS under assumptions of C and $ D2 $QF scenarios. We evaluate the 90%-CL upper limits on the CEνNS amplitude that is 4.3/3.1 times larger than the Standard-Model prediction for C/$ D2 $ based on the statistical analysis of the data. Under the assumption of the most optimistic QF scenario $ D1 $, the data demonstrate a $ 2.5\; \sigma $ tension with the Standard Model CEνNS, indicating either the presence of non-standard interactions or a bias in the QF model. This latter discrepancy is softened by the systematic uncertainty of the energy scale calibration. It should be mentioned that our current analysis is based on the direct comparison of the count rates during reactor-ON and reactor-OFF regimes, which does not require the simulation of contributions of the background with additional assumptions of its shapes and level.

The projected sensitivity, estimated based on the measured background count rate, shows that 3-σ CEνNS could be detected within a few years of stable data if a background model is used. Tests aimed at reducing the background and improving the energy threshold of the HPGe detector were conducted under laboratory conditions at the Joint Institute for Nuclear Research (JINR).

The authors are grateful to the KNPP directorate and staff for extensive support and practical help in performing measurements at the reactor site. The authors are thankful to Anton Lukyashin (MEPhI, MIREA) and Valery Sinev (INR RAS, MEPhI) for discussions about the reactor antineutrino energy spectra.